L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s − 8-s + 9-s + 10-s − 6.24·11-s + 12-s + 5.65·13-s − 15-s + 16-s − 5.41·17-s − 18-s + 1.17·19-s − 20-s + 6.24·22-s + 8.82·23-s − 24-s + 25-s − 5.65·26-s + 27-s + 5.41·29-s + 30-s + 1.75·31-s − 32-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.447·5-s − 0.408·6-s − 0.353·8-s + 0.333·9-s + 0.316·10-s − 1.88·11-s + 0.288·12-s + 1.56·13-s − 0.258·15-s + 0.250·16-s − 1.31·17-s − 0.235·18-s + 0.268·19-s − 0.223·20-s + 1.33·22-s + 1.84·23-s − 0.204·24-s + 0.200·25-s − 1.10·26-s + 0.192·27-s + 1.00·29-s + 0.182·30-s + 0.315·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.292201301\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.292201301\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + 6.24T + 11T^{2} \) |
| 13 | \( 1 - 5.65T + 13T^{2} \) |
| 17 | \( 1 + 5.41T + 17T^{2} \) |
| 19 | \( 1 - 1.17T + 19T^{2} \) |
| 23 | \( 1 - 8.82T + 23T^{2} \) |
| 29 | \( 1 - 5.41T + 29T^{2} \) |
| 31 | \( 1 - 1.75T + 31T^{2} \) |
| 37 | \( 1 - 8.24T + 37T^{2} \) |
| 41 | \( 1 - 6.48T + 41T^{2} \) |
| 43 | \( 1 + 5.07T + 43T^{2} \) |
| 47 | \( 1 + 6.24T + 47T^{2} \) |
| 53 | \( 1 - 11.6T + 53T^{2} \) |
| 59 | \( 1 - 8.82T + 59T^{2} \) |
| 61 | \( 1 - 0.343T + 61T^{2} \) |
| 67 | \( 1 + 5.07T + 67T^{2} \) |
| 71 | \( 1 - 12.4T + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 10.8T + 83T^{2} \) |
| 89 | \( 1 + 4.82T + 89T^{2} \) |
| 97 | \( 1 + 10.4T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.359389671984140178074224848775, −8.449848927606038986166160857196, −8.255652044158678091899951984206, −7.26040578021397072339642465224, −6.52779371903591659927446663777, −5.37612702172934133440579256508, −4.36440397428401516971861550638, −3.14514599817664623456611992581, −2.43280695826534350118871297251, −0.881882603000887135653074500524,
0.881882603000887135653074500524, 2.43280695826534350118871297251, 3.14514599817664623456611992581, 4.36440397428401516971861550638, 5.37612702172934133440579256508, 6.52779371903591659927446663777, 7.26040578021397072339642465224, 8.255652044158678091899951984206, 8.449848927606038986166160857196, 9.359389671984140178074224848775